Eulerian Reference Frame Research Articles

A lamellar model for analysis of liquid-liquid mixing

Two quantities, a ν, intermaterial area per unit volume and s, striation thickness, are defined to describe the state of a system mixed or being mixed. These quantities are related to one another and to the fluid mechanics of mixing in Lagrangian and Eulerian frames of reference. The model considers that mechanical mixing is synonymous with deformation. An equation for rate of generation of intermaterial area is presented and integrated for systems of engineering interest. An upper bound for this rate of generation is obtained for the case of Newtonian incompressible fluids. Special attention is given to the microscale of mixing where reaction and diffusion occur. It is shown that diffusion and reaction can be combined into a single variable due to a particular choice of reference frame. In this way the necessary tools for analyzing liquid-liquid mixing are presented. The restrictions to this analysis are: (i) microscale can be represented by a lamellar structure, (ii) no surface tension effects, (iii) no instabilities at the microscale of mixing.

On the dispersion of small particles suspended in an isotropic turbulent fluid

A solution to the dispersion of small particles suspended in a turbulent fluid is presented, based on the approximation proposed by Phythian for the dispersion of fluid points in an incompressible random fluid. Motion is considered in a frame moving with the mean velocity of the fluid, the forces acting on the particle being taken as gravity and a fluid drag assumed linear in the particle velocity relative to that of the fluid. The probability distribution of the fluid velocity field in this frame is taken as Gaussian, homogeneous, isotropic, stationary and of zero mean. It is shown that, in the absence of gravity, the long-time particle diffusion coefficient is in general greater than that of the fluid, approaching with increasing particle relaxation time a value consistent with the particle being in an Eulerian frame of reference. The effect of gravity is consistent with Yudine's effect of crossing trajectories, reducing unequally the particle diffusion in directions normal to and parallel to the direction of the gravitational field. To characterize the effect of flow and gravity on particle diffusion it has been found useful to use a Froude number defined in terms of the turbulent intensity rather than the mean velocity. Depending upon the value of this number, it is found that the particle integral time scale may initially decrease with increasing particle relaxation time though it eventually rises and approaches the particle relaxation time. It is finally shown how this analysis may be extended to include the extra forces generated by the fluid and particle accelerations.

Lagrangian coordinate transformations in self-similar supernova blast-wave models

The Sedov self-similar solution of the fluid equations provides a description of a spherical supernova blast-wave during its adiabatic phase in a stationary (Eulerian) reference frame. We provide an exact transformation to a Lagrangian coordinate system, a non-inertial frame of reference which follows each individual gas particle. The physical properties of the gas in this comoving reference system are evaluated as a function of time. Numerical methods to facilitate Eulerian-Lagrangian and Lagrangian-Eulerian coordinate transformations are provided.

Finite elasto-plastic deformation—I: Theory and numerical examples

It is demonstrated that the problem of elasto-plastic finite deformation is governed by a quasi-linear model irrespective of deformation magnitude. This feature follows from the adoption of a rate viewpoint toward finite deformation analysis in an Eulerian reference frame. Objectivity of the formulation is preserved by introduction of a frame-invariant stress rate.Equations for piece-wise linear incremental finite element analysis are developed by application of the Galerkin method to the instantaneously linear governing differential equations of the quasi-linear model. Numerical solution capability has been established for problems of plane strain and plane stress. The accuracy of the numerical analysis is demonstrated by consideration of a number of problems of homogeneous finite deformation admiting comparative analytic solution. It is shown that accurate, objective numerical solutions can be obtained for problems involving dimensional changes of an order of magnitude and rotations of a full radian.

A vortex model to relate eulerian and lagrangian “ turbulent” velocity fields

AbstractThe phenomenon of turbulent diffusion was studied by using computed models. Free pseudo‐turbulent fluid flow was simulated on a digital computer by ensembles of randomly positioned moving and interacting vortices to represent the eddy structure of the turbulence. A vortex model, which had statistical values corresponding to a measured real flow in an Eulerian frame of reference, was then obtained. By considering the motion of a particular vortex or a fluid particle in the computed pseudo‐turbulent flow it was possible to determine measurements in a Lagrangian frame of reference. Estimates of the mean square lateral excursions of a marked particle were obtained and finally the Lagrangian auto correlation was compared with the auto correlation in a moving frame of reference.

On the oscillatory motions of translating elastic cables

The equations of in-plane motion for an elastic catenary translating uniformly between its end supports are derived in what is, essentially, an Eulerian frame of reference. These equations are solved analytically, albeit approximately, for the case where the catenary is shallow and the tension dominated by the cable section modulus. Computed solutions for the natural frequencies and modes of a catenary of sag-span ratio 1:20 are presented and the modal characteristics are shown to be of an unusual form involving phase disparities from point to point on the cable as the cable oscillates at a natural frequency.

Spatial structure in the viscous sublayer

An experimental investigation was performed to study the spatial coherence of structures in the sublayer of a turbulent boundary layer observed previously by flow visualization. The present work verifies these observations in an Eulerian reference frame and develops a statistical description of the phenomenon. The technique involves simultaneous digital sampling of an array of constant temperature hot-wire anemometers arranged to extract information about a spanwise variation in flow quantities. The quantitative description agrees with dimensionless measures of the structure scales previously published.

Water waves measured with a laser flowmeter

The laboratory measurement of velocity fields under water waves, which is fundamental to the study of many coastal and oceanographic phenomena, has up until now been humpered by lack of satisfactory instrumentation. Photographing of smail floats or tracer particles has been used extensively to measure the fluid motion in a Langrangian manner (e.g. Russell and Osorio 1957), but the analysis of data is extremely tedious and furthermore, for engineering purposes, we are more often interested in measuring to an Eulerian frame of reference. For this purpose prope!ler meters have been used and these can be made as small as 5 mm in diameter and capable of registering down to about 0.5 cm! s. The problem here is that their response to a sudden change in velocity is delayed by the inertia of the propeller and, furthermore, several revolutions must be recorded before an accurate velocity reading is obtained, meaning that it is impossible to measure instantaneous velocities with any accuracy. Although hot film anemometry is in principle capable of overcoming these limitations, it has never become a practical system for water wave measurements because of the problems of calibration drift when using contaminated water. In a study of wave velocities near the bed, Sleath (1970) has usec1 a novel device consisting of a fine fibreglass wire stretched across the channel. The force on the wire at any instant is then recorded and is a measure of the water velocity. This ovcrcomes the difficuIty of responsc timc but cannot easily be used to produce detailed information on velocity fields as firstly it is not